If 8.40 kJ of heat is needed to raise the temperature of a sample of metal from 15 °C to 20 °C, how many kilojoules of heat will be required to raise the temperature of the same sample of metal from 25 °C to 40 °C?

1 Answer
Dec 29, 2017

Answer:

#"25 kJ"#

Explanation:

The trick here is to realize that because the sample of metal has the same mass in both cases, you can say that

#q_2 = (DeltaT_2)/(DeltaT_1) * q_1#

Here

  • #q_1# is the amount of heat needed to raise the temperature of the sample by #DeltaT_1 = 20^@"C" - 15^@"C"#
  • #q_2# is the amount of heat needed to raise the temperature of the sample by #DeltaT_2 = 40^@"C" - 25^@"C"#

This equation can be found by using the fact that the heat absorbed by the metal can be calculated using the equation

#color(blue)(ul(color(black)(q = m * c * DeltaT)))#

Here

  • #m# is the mass of the sample
  • #c# is the specific heat of the metal

In your case, you can say that

#q_1 = m * c * DeltaT_1#

and

#q_2 = m * c * DeltaT_2#

Divide these two equations

#q_1/q_2 = (color(red)(cancel(color(black)(m * c))) * DeltaT_1)/(color(red)(cancel(color(black)(m * c))) * DeltaT_2)#

to get

#q_2 = (DeltaT_2)/(DeltaT_1) * q_1#

This equation tells you that in order to increase the temperature of the metal by a factor #(DeltaT_2)/(DeltaT_1)# when the mass of the metal is constant, the amount of heat supplied must also increase by a factor of #(DeltaT_2)/(DeltaT_1)#.

So, plug in your values to find

#q_2 = ((40 - 25) color(red)(cancel(color(black)(""^@"C"))))/((20-15)color(red)(cancel(color(black)(""^@"C")))) * "8.40 kJ"#

#color(darkgreen)(ul(color(black)(q_2 = "25 kJ")))#

The answer is rounded to two sig figs.